She wasn’t conscripted, she was elected to perform an administrative duty which by necessity must be performed impartially, this necessity was understood when the job was accepted.
However, the long standing definition of marriage has been altered. This is unexpected and does pose a religious liberty problem, she may never have sought the position had this new and radical definition been in place. Despite that, I think it is going to be very difficult to prevail with a religious liberty argument.
I think another more straightforward argument is to reject the illegal order of the court.
Haven't they questioned the legality of the order in their appeals and have gotten nowhere?